Riesz-Markov-Kakutani Theorem: Total variation Norm and weak*

Question

Let $X$ be a compact Hausdorff space, the Riesz-Markov-Kakutani theorem states that the topological dual of $C(X)$ is the space $M(X)$ of regular countably additive complex Borel measure on $X$ equipped with the weak* topology.
It also states that given $\mu \in M(X)$ as linear functional its norm is equal to the total-variation norm of $\mu$ as complex measure.

Does this mean that the topology induced by the total-variation norm coincides with the weak* topology?

Answer 1

The dual of $C(X)$ is $M(X)$ with the total variation norm. You are quoting RMK theorem wrongly.